The following is from Descartes’s method (1637) in which he provides four rules
ID: 3122347 • Letter: T
Question
The following is from Descartes’s method (1637) in which he provides four rules for his philosophical method not be violated “even in a single instance”:
The first rule was never to accept anything as true unless I recognized it to be certainly and evidently such: that is, carefully to avoid all precipitation and prejudgement, and to include nothing in my conclusions unless it presented itself so clearly and distinctly to my mind that there was no reason or occasion to doubt it.
The second was to divide each of the difficulties which I encountered into as many parts as possible, and as might be required for an easier solution.
The third was to think in an orderly fashion when concerned with the search for truth, beginning with the things which were simplest and easiest to understand, and gradu- ally and by degrees reaching toward more complex knowledge, even treating, as though ordered, materials which were not necessarily so.
The last was, both in the process of searching and in reviewing when in difficulty, always to make enumerations so complete, and reviews so general, that I would be certain that nothing was omitted.
Descartes intended his G eom etrie as one of the applications of this method. How do the following excerpts, support his method? How might Descartes’ method work outside of mathematics (or is it too mathematical to work in other domains)?
If, then, we wish to solve any problem, we first suppose the solution already effected, and give names to all the lines that seem needful for its construction–to those that are unknown as well as to those that are known. Then, making no distinction between known and unknown lines, we must unravel the difficulty in any way that shows most naturally the relations between these lines, until we find it possible to express a single quantity in two ways. This will constitute an equation, since the terms of one of these two expressions are together equal to the terms of the other.
We must find as many such equations as there are supposed to be unknown lines; but if, after considering everything involved, so many cannot be found, it is evident that the question is not entirely determined. In such a case we may choose arbitrarily lines of known length for each unknown line to which there corresponds no equation.
If there are several equations, we must use each in order, either considering it alone or comparing it with others, so as to obtain a value for each of the unknown lines; and so we must combine them until there remains a single unknown line which is equal to some known line, or whose square, cube, fourth power, fifth power, sixth power, etc., is equal to the sum or difference of two or more quantities, one of which is known, while the others consist of mean proportionals between unity and this square, or cube, or fourth power, etc., multiplied by other known lines.
Thus, all the unknown quantities may be expressed in terms of a single quantity, whenever the problem can be constructed by means of circles and straight lines, or by conic sections, or even by some other curve of degree not greater than the third or fourth.
But I shall not stop to explain this in more detail, because I should deprive you of the pleasure of mastering it yourself, as well as of the advantage of training your mind by working over it, which is in my opinion the principal benefit to be derived from this science. Because, I find nothing here so difficult that it cannot be worked out by any one at all familiar with ordinary geometry and with algebra, who will consider carefully all that is set forth in this treatise.
Explanation / Answer
Gaussian elimination adapts method. That is, if M equations in n unknowns are there, then first we eliminate x1,(say), from all but 1st equation, assuming that x1 is present in the 1st equation. If not, then we exchange the equations 1st with another which contains x1. Now, x1 is present only in the 1st equation. Now locate x2, if x2 is present in one of the equations from 2nd to M -th then making exchange( with one of these equations ) x2 can be brought to 2nd equation and with its help x2 can be eliminated from 3rd to M -th equation. Proceeding this way we find that r variables can be eliminated where r is the rank of the matrix. The right side values pertaining to these r equations may have any values. However, if there are other equations in which the right side value is nonzero, then the system is Iinconsistent. Otherwise, the system is consistent, in which it can be solved by back substituion.
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